An edge crack and a crack close to the vertex of a wedge

Abstract

Two model problems of an elastic wedge with an internal and edge crack are analyzed. The problem of an internal crack reduces to an order-4 vector Riemann-Hilbert problem whose matrix kernel entries are meromorphic functions and have exponential factors. When the internal crack is located along one of the wedge sides, an efficient method of solution is proposed. It requires a factorization of the order-2 matrix coefficient associated with the corresponding problem of an edge crack and the solution of an infinite system of linear algebraic system with an exponential rate of convergence of an approximate solution to the exact one. The order-2 Khrapkov's factorization is modified by splitting the matrix kernel into a scalar dominant function and a ``regular" matrix whose factorization is more convenient for numerical purposes. Expressions for the stress intensity coefficients and the potential energy released when the crack advances are derived. Asymptotic relations for the stress intensity coefficients and the potential energy when one of the crack tips is close the wedge vertex are obtained.

Figures (6)

• Figure 1: Stress intensity factors $K_I$ and $K_{II}$ versus the angle $\alpha$ measured in degrees when $b=1$ and $P_1=P_2=1$.
• Figure 2: Stress intensity factors $K_I$ and $K_{II}$ versus the angle $\alpha$ measured in degrees when $b=1$ and the loads are constant, $P_1=1$ and $P_2=0$.
• Figure 3: Stress intensity factors $K_I$ and $K_{II}$ versus the angle $\alpha$ when $b=1$, $k^\circ_\theta=1$, and the loads correspond to the first eigen-solution: $p_\theta(r)=r^{\mu-1}$, $p_{r\theta}=k_* r^{\mu-1}$.
• Figure 4: The stress intensity factors $K_I$ and $K_{II}$ versus the angle $\alpha$ when $b=1$, $k^\circ_\theta=1$, and the loads correspond to the second eigen-solution are $p_\theta(r)=r^{\mu_0-1}$, $p_{r\theta}=k_* r^{\mu_0-1}$.
• Figure 5: Entries $c_q=\cos 2q$ and $c_s=\sin 2q$ of the matrix $Q$ versus the angle $\alpha$.